Monotone Gain, First-Order Autocorrelation and Zero-Crossing Rate

Abstract

The effect of a linear filter with monotone gain on the first-order autocorrelation of a weakly stationary time series is discussed. When the gain is monotone increasing, the first-order autocorrelation cannot increase. Otherwise, when the gain is monotone decreasing, the correlation cannot decrease. Further, when the gain is strictly monotone, the first-order autocorrelation is unchanged if and only if the process is a pure sinusoid with probability 1. Under the Gaussian assumption, the zero-crossing rate moves oppositely from the first-order autocorrelation.